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Smoothed nonparametric tests and their properties

Yoshihiko Maesono, Taku Moriyama, Mengxin Lu

Abstract

In this paper we propose new smoothed sign and Wilcoxon's signed rank tests, which are based on a kernel estimator of the underlying distribution function of data. We discuss approximations of $p$-values and asymptotic properties of these tests. The new smoothed tests are equivalent to the ordinary sign and Wilcoxon's tests in the sense of the Pitman's asymptotic relative efficiency, and the differences of the ordinary and the new tests converge to zero in probability. Under the null hypothesis, the main terms of the asymptotic expectations and variances of the tests do not depend on the underlying distribution. Though the smoothed tests are not distribution-free, we can obtain Edgeworth expansions with residual term $o(n^{-1})$, which do not depend on the underlying distribution.

Smoothed nonparametric tests and their properties

Abstract

In this paper we propose new smoothed sign and Wilcoxon's signed rank tests, which are based on a kernel estimator of the underlying distribution function of data. We discuss approximations of -values and asymptotic properties of these tests. The new smoothed tests are equivalent to the ordinary sign and Wilcoxon's tests in the sense of the Pitman's asymptotic relative efficiency, and the differences of the ordinary and the new tests converge to zero in probability. Under the null hypothesis, the main terms of the asymptotic expectations and variances of the tests do not depend on the underlying distribution. Though the smoothed tests are not distribution-free, we can obtain Edgeworth expansions with residual term , which do not depend on the underlying distribution.

Paper Structure

This paper contains 5 sections, 9 theorems, 112 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Asymptotic properties
  3. Smoothed Wilcoxon's signed rank test
  4. Simulation study
  5. Appendices

Key Result

Theorem 1

Let us assume that $f'$ exists and is continuous at a neighborhood of $-\theta$, and $h_n=cn^{-d} (c>0, \frac{1}{4}<d<\frac{1}{2})$. If the kernel $k(\cdot)$ is symmetric around the origin and the standardized $\widetilde{S}$ is asymptotically normal, that is in law.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9